Subgroups — C₃₁×#17787 #21474546

Order 7936.

778 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1565	C₂	1	8	yes	maximal, complemented
4	C₃₁×#1564	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₃₁	C₄	1	1	yes	characteristic
6	C₂²×M₁₆×C₃₁	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#431	—	2	2	no	—
9	C₃₁×#431	—	2	2	no	—
10	C₃₁×#431	C₂²	1	4	yes	—
11	C₃₁×#431	C₂²	1	4	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
14	C₄×M₁₆×C₃₁	C₄	1	4	yes	—
15	C₄×M₁₆×C₃₁	C₂²	1	4	yes	—
16	C₂×C₃₁×#97	C₄	1	2	yes	—
17	C₂×C₄×C₈×C₃₁	—	2	2	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#335	C₂²	1	8	yes	complemented
20	C₃₁×#334	C₂²	1	8	yes	complemented
21	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
24	C₂×M₁₆×C₃₁	—	2	4	no	—
25	C₂×M₁₆×C₃₁	C₈	1	8	yes	complemented
26	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
27	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₃₁	—	2	4	no	—
29	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	D₄	1	2	yes	—
35	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
37	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×C₄⋊C₈	—	2	4	no	—
40	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
41	C₃₁×C₂²⋊C₈	—	2	4	no	—
42	C₃₁×C₂²⋊C₈	—	2	4	no	—
43	C₃₁×#97	C₂×C₄	1	8	yes	—
44	C₄×C₈×C₃₁	—	2	4	no	—
45	C₄×C₈×C₃₁	—	2	4	no	—
46	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
47	C₄×C₈×C₃₁	C₂³	1	4	yes	—
48	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	—	4	2	no	—
50	C₂²×C₄×C₃₁	—	4	2	no	—
51	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
59	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
65	M₁₆×C₃₁	—	2	8	no	—
66	M₁₆×C₃₁	C₂×C₈	1	16	yes	complemented
67	C₂×C₈×C₃₁	—	2	4	no	—
68	C₂×C₈×C₃₁	—	2	8	no	—
69	C₂×C₈×C₃₁	—	4	8	no	—
70	C₂×C₈×C₃₁	C₂×C₈	1	8	yes	complemented
71	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
73	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
74	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
75	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
76	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
78	C₄²×C₃₁	C₂×D₄	1	2	yes	—
79	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
81	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
82	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
83	#17787	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₃₁	—	4	1	no	—
85	C₂³×C₃₁	—	4	1	no	—
86	C₂³×C₃₁	C₄⋊C₈	1	4	yes	—
87	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
88	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	—	2	4	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	4	1	no	—
94	C₂×C₄×C₃₁	—	4	2	no	—
95	C₂×C₄×C₃₁	—	4	1	no	—
96	C₂×C₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
98	C₂×C₄×C₃₁	C₄⋊C₈	1	8	yes	—
99	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
100	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
107	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
109	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
112	C₈×C₃₁	—	2	4	no	—
113	C₈×C₃₁	—	4	8	no	—
114	C₈×C₃₁	C₂²×C₈	1	8	yes	—
115	C₂×#431	C₂×C₃₁	1	2	yes	—
116	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
117	#1565	C₂×C₃₁	1	8	yes	complemented
118	#1564	C₂×C₃₁	1	4	yes	complemented
119	C₂²×C₃₁	—	2	2	no	—
120	C₂²×C₃₁	—	2	1	no	—
121	C₂²×C₃₁	—	2	2	no	—
122	C₂²×C₃₁	—	2	1	no	—
123	C₂²×C₃₁	—	4	4	no	—
124	C₂²×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
125	C₂²×C₃₁	C₂×C₄⋊C₈	1	4	yes	—
126	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	2	2	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	C₂×C₄⋊C₈	1	4	yes	—
132	C₄×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
133	C₄×C₃₁	#430	1	2	yes	—
134	C₄×C₃₁	C₂³×C₈	1	1	yes	characteristic
135	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
136	C₄×C₃₁	#575	1	2	yes	—
137	C₂²×M₁₆	C₄×C₃₁	1	1	yes	characteristic
138	C₂²×M₁₆	C₄×C₃₁	1	1	yes	characteristic
139	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
140	#431	—	2	2	no	—
141	#431	—	2	2	no	—
142	#431	C₂²×C₃₁	1	4	yes	—
143	#431	C₂²×C₃₁	1	4	yes	—
144	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
145	C₂×C₂²⋊C₈	—	2	2	no	—
146	C₄×M₁₆	C₄×C₃₁	1	4	yes	—
147	C₄×M₁₆	C₂²×C₃₁	1	4	yes	—
148	C₂×#97	C₄×C₃₁	1	2	yes	—
149	C₂×C₄×C₈	—	2	2	no	—
150	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
151	#335	C₂²×C₃₁	1	8	yes	complemented
152	#334	C₂²×C₃₁	1	8	yes	complemented
153	C₂×C₃₁	—	2	4	no	—
154	C₂×C₃₁	—	2	2	no	—
155	C₂×C₃₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
156	C₂×C₃₁	#2911	1	1	yes	characteristic
157	C₂×C₃₁	C₂×#430	1	1	yes	characteristic
158	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
159	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
160	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
161	C₂×M₁₆	—	2	4	no	—
162	C₂×M₁₆	C₈×C₃₁	1	8	yes	complemented
163	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
164	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
165	C₂²×C₈	—	2	4	no	—
166	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
167	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
168	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
169	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
170	C₂×C₄²	D₄×C₃₁	1	2	yes	—
171	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
172	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
173	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
174	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
175	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
176	C₄⋊C₈	—	2	4	no	—
177	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
178	C₂²⋊C₈	—	2	4	no	—
179	C₂²⋊C₈	—	2	4	no	—
180	#97	C₂×C₄×C₃₁	1	8	yes	—
181	C₄×C₈	—	2	4	no	—
182	C₄×C₈	—	2	4	no	—
183	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
184	C₄×C₈	C₂³×C₃₁	1	4	yes	—
185	C₃₁	#17787	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
190	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
193	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
194	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
196	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
197	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
199	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
201	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
203	M₁₆	—	2	8	no	—
204	M₁₆	C₂×C₈×C₃₁	1	16	yes	complemented
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	8	no	—
207	C₂×C₈	—	4	8	no	—
208	C₂×C₈	C₂×C₈×C₃₁	1	8	yes	complemented
209	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
210	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
211	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
212	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
213	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
214	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
215	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
216	C₄²	C₂×D₄×C₃₁	1	2	yes	—
217	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
219	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
220	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
221	C₂³	—	4	1	no	—
222	C₂³	—	4	1	no	—
223	C₂³	C₃₁×C₄⋊C₈	1	4	yes	—
224	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
225	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	C₃₁×C₄⋊C₈	1	8	yes	—
235	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
236	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
237	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
239	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
240	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
243	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
244	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
245	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
246	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
249	C₈	—	2	4	no	—
250	C₈	—	4	8	no	—
251	C₈	C₂²×C₈×C₃₁	1	8	yes	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	4	no	—
257	C₂²	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
258	C₂²	C₂×C₃₁×C₄⋊C₈	1	4	yes	—
259	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	1	no	—
264	C₄	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
265	C₄	C₂×C₃₁×C₄⋊C₈	1	4	yes	—
266	C₄	C₃₁×#430	1	2	yes	—
267	C₄	C₂³×C₈×C₃₁	1	1	yes	characteristic
268	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
269	C₄	C₃₁×#575	1	2	yes	—
270	C₂	—	2	4	no	—
271	C₂	—	2	2	no	—
272	C₂	C₂²×C₃₁×C₄⋊C₈	1	1	yes	characteristic
273	C₂	C₃₁×#2911	1	1	yes	characteristic
274	C₂	C₂×C₃₁×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₄²×C₃₁	C₂×D₄
commutator	C₄	C₂³×C₈×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₃₁×#1565	C₂	semidirect
4. C₃₁×#1564	C₂	semidirect
19. C₃₁×#335	C₂²	semidirect
20. C₃₁×#334	C₂²	semidirect
25. C₂×M₁₆×C₃₁	C₈	semidirect
66. M₁₆×C₃₁	C₂×C₈	semidirect
70. C₂×C₈×C₃₁	C₂×C₈	semidirect
83. #17787	C₃₁	direct (N × H)
117. #1565	C₂×C₃₁	semidirect
118. #1564	C₂×C₃₁	semidirect
151. #335	C₂²×C₃₁	semidirect
152. #334	C₂²×C₃₁	semidirect
162. C₂×M₁₆	C₈×C₃₁	semidirect
185. C₃₁	#17787	direct (N × H)
204. M₁₆	C₂×C₈×C₃₁	semidirect
208. C₂×C₈	C₂×C₈×C₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#431	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#1565	2
C₃₁×#1564	2
#17787	31

Subgroups — C31×#17787 #21474546

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17787 #21474546